ajk...the Holy See is the Government of the Vatican City State. A bit like the RUssian Government is the head of the the Russian Commonwealth of Independent States.
Vatican State is a theocracy, as I said. The Holy See is its government (unless it has a civil parliament full of elected Swiss Guards I don't know about).
In theocracies the religious leaders do civil/ secular business. That is the definition of a theocracy; think of the Dalai Llama. He does a LOT of secular business.
No, the Holy See is the juridical and sovereign entity that embodies the Catholic Church. By legal fiction agreed upon by the international community and by the member-states of the United Nations (successor orgnization to the League of Nations), the Holy See is deemed also a member-sate.
The Holy See has been in existence as such since time immemorial (at least in the Middle Ages) and states (and countries) have been in diplomatic relations with her. The Vatican City came into being only in 1929 when the Lateran Treaty made it the independent territory of the Holy See.
The day-to-day governance of the Holy See is undertaken by the Roman Curia led by the Secretariat of State. The Curia functions at the behest of the Pope.